FPT Approximations for Connected Maximum Coverage

TL;DR

The paper addresses connectivity-constrained coverage by formalizing a unified model, PartialConRBDS, that jointly considers a connectivity graph $G_{ ext{conn}}$ on red elements and a coverage incidence graph $G_{ ext{cov}}=(R\cup B,E)$. It develops two FPT-approximation schemes for the case where the coverage graph is $K_{d,d}$-free: a $(1,1-\varepsilon)$-approximation with time $2^{\mathcal{O}(k^2 d/\varepsilon)} \cdot |\mathcal{I}|^{\mathcal{O}(1)}$ and a $(1+\varepsilon,1)$-approximation with time $2^{\mathcal{O}(kd(k^2+\log d))} \cdot |\mathcal{I}|^{\mathcal{O}(1/\varepsilon)}$, independent of $G_{ ext{conn}}$, aided by neighborhood-sparsifier techniques. The work also delineates hardness boundaries via W[1]/W[2] reductions and ETH-based lower bounds, showing that the proposed $K_{d,d}$-free restrictions yield tractable FPT-approximation regimes. Finally, it proposes a dichotomy-program framework to classify problems by connectivity- and incidence-graph families, outlining a path to map the frontier between tractable and hard regimes for connectivity-constrained coverage. These results advance the understanding of when connectivity constraints can be reconciled with near-optimal coverage in an FPT setting.

Abstract

We revisit connectivity-constrained coverage through a unifying model, Partial Connected Red-Blue Dominating Set. Given a red-blue bipartite graph $G$ and an auxiliary connectivity graph $G_{conn}$ on red vertices, and integers $k, t$, the task is to find a $k$-sized subset of red vertices that dominates at least $t$ blue vertices, and that induces a connected subgraph in $G_{conn}$. This formulation captures connected variants of Max Coverage, Partial Dominating Set, and Partial Vertex Cover studied in prior literature. After identifying (parameterized) inapproximability results inherited from known problems, we first show that the problem is fixed-parameter tractable by $t$. Furthermore, when the bipartite graph excludes $K_{d,d}$ as a subgraph, we design (resp. efficient) parameterized approximation schemes for approximating $t$ (resp. $k$). Notably, these FPT approximations do not impose any restrictions on $G_{conn}$. Together, these results chart the boundary between hardness and FPT-approximability for connectivity-constrained coverage.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Lemma 3: Neighborhood Sparsifiers for $K_{d,d}$-free coverage